Question

In Example \(7.9\) on page \(305\), we conducted a simulation to check the plausibility of the central limit theorem. The variable under consideration there is household size, and the population consists of all U.S households. A frequency distribution for household size of U.S households is presented in Table \(7.7\).

a. Suppose that you simulate \(1000\) samples of four households each, determine the sample mean of each of the \(1000\) samples, and obtain a histogram of the \(1000\) sample means. Would you expect the histogram to be bell shaped? Explain your answer.

b. Carry out the tasks in part (a) and note the shape of the histogram.

c. Repeat Parts (a) and (b) for sample of size \(10\).

d. Repeat Parts (a) and (b) for sample of size \(100\).

Short answer

Part a. No.

Part b. Strongly skewed

Part c. Strongly skewed

Part d. Roughly Bell Shaped.

Step-by-step solution

Part a. Step 1. Given information

Part a. Step 2. Explanation

\(n=Sample size=4\)

Randomly generate \(4\) integers from \(1\) to \(1160\)

This can be done using the command \(randInt(1,1160,4)\) while using a \(T183/84\) - calculator

While the value is between \(1,311\) , in that case the household consists of \(x=1\) people

While the value is between \(312,697\), in that case the household consists of \(x=2\) people

While the value is between \(698,885\), in that case the household consists of \(x=3\) people

While the value is between \(886,1047\), in that case the household consists of \(x=4\) people

While the value is between \(1048,1119\), in that case the household consists of \(x=5\) people

While the value is between \(1129,1146\), in that case the household consists of \(x=6\) people

While the value is between \(1147,1160\), in that case the household consists of \(x=7\) people

After this, calculate the sample mean for the above sample.

Repeat the process \(1000\) times and the histogram needs to be created for the sample means.

As the sample size is four, the sample size is small and the central limit theorem is not applicable, the histogram is expected to display the similar distribution to the population and therefore, the distribution is expected to be strongly skewed because the population is also skewed. This states the resulting histogram is not bell-shaped.

Part b. Step 1. Explanation

\(n=Sample size=4\)

Randomly generate \(4\) integers from \(1\) to \(1160\)

This can be done using the command \(randInt(1,1160,4)\) while using a \(T183/84\) - calculator

While the value is between \(1,311\) , in that case the household consists of \(x=1\) people

While the value is between \(312,697\), in that case the household consists of \(x=2\) people

While the value is between \(698,885\), in that case the household consists of \(x=3\) people

While the value is between \(886,1047\), in that case the household consists of \(x=4\) people

While the value is between \(1048,1119\), in that case the household consists of \(x=5\) people

While the value is between \(1129,1146\), in that case the household consists of \(x=6\) people

While the value is between \(1147,1160\), in that case the household consists of \(x=7\) people

After this, calculate the sample mean for the above sample.

Repeat the process \(1000\) times and the histogram needs to be created for the sample means.

As the sample size is four, the sample size is small and the central limit theorem is not applicable, the histogram is expected to display the similar distribution to the population and therefore, the distribution is expected to be strongly skewed because the population is also skewed. This states the resulting histogram is not bell-shaped.

Part c. Step 1. Explanation

\(n=Sample size=10\)

Randomly generate \(4\) integers from \(1\) to \(1160\)

This can be done using the command \(randInt(1,1160,4)\) while using a \(T183/84\) - calculator

While the value is between \(1,311\) , in that case the household consists of \(x=1\) people

While the value is between \(312,697\), in that case the household consists of \(x=2\) people

While the value is between \(698,885\), in that case the household consists of \(x=3\) people

While the value is between \(886,1047\), in that case the household consists of \(x=4\) people

While the value is between \(1048,1119\), in that case the household consists of \(x=5\) people

While the value is between \(1129,1146\), in that case the household consists of \(x=6\) people

While the value is between \(1147,1160\), in that case the household consists of \(x=7\) people

After this, calculate the sample mean for the above sample.

Repeat the process \(1000\) times and the histogram needs to be created for the sample means.

As the sample size is ten, the sample size is small and the central limit theorem is not applicable, the histogram is expected to display the similar distribution to the population and therefore, the distribution is expected to be strongly skewed because the population is also skewed. This states the resulting histogram is not bell-shaped.

Part d. Step 1. Explanation

\(n=Sample size=100\)

Randomly generate \(4\) integers from \(1\) to \(1160\)

This can be done using the command \(randInt(1,1160,4)\) while using a \(T183/84\) - calculator

While the value is between \(1,311\) , in that case the household consists of \(x=1\) people

While the value is between \(312,697\), in that case the household consists of \(x=2\) people

While the value is between \(698,885\), in that case the household consists of \(x=3\) people

While the value is between \(886,1047\), in that case the household consists of \(x=4\) people

While the value is between \(1048,1119\), in that case the household consists of \(x=5\) people

While the value is between \(1129,1146\), in that case the household consists of \(x=6\) people

While the value is between \(1147,1160\), in that case the household consists of \(x=7\) people

After this, calculate the sample mean for the above sample.

Repeat the process \(1000\) times and the histogram needs to be created for the sample means.

As the sample size is hundred, the sample size is large and the central limit theorem is applicable, the histogram is expected to display a rough normal distribution. This states the resulting histogram is roughly bell-shaped.